U.S. patent application number 11/995808 was filed with the patent office on 2009-02-19 for model set adaptation by probability mass diffusion.
Invention is credited to Thomas Kronhamn.
Application Number | 20090048811 11/995808 |
Document ID | / |
Family ID | 35929801 |
Filed Date | 2009-02-19 |
United States Patent
Application |
20090048811 |
Kind Code |
A1 |
Kronhamn; Thomas |
February 19, 2009 |
MODEL SET ADAPTATION BY PROBABILITY MASS DIFFUSION
Abstract
A method of performing a sequence of measurements, z, R; M;
(t.sub.1,t.sub.2), of at least one parameter and recursively
performing predictions. The method comprising the steps of--based
on at least on a first measurement instance (M (t.sub.k); (k)),
predicting the outcome (x, P) for at least two models (C, S);
--after a subsequent measurement instance (M
(t.sub.k+T.sub.p)(k+Tp)) updating the models (C, S) for the
corresponding point in time, whereby the prediction made on the
basis of the first measurement instance is updated in the light of
the subsequent measurement instance; and--re-arranging at least one
model (C, S) for the subsequent measurement instance
(t.sub.k+T.sub.p) (k+Tp), whereby one updated model influences
another updated model. For a model set comprising at least one
complementary (C) model and at least one sub (S) model, under the
step of rearranging the S model never influences the C model. For a
model set comprising exclusively complementary (L, N, R) models,
under the step of re-arranging, for a given pair of models within
the model set (L, N, R), a model having a higher probability (.mu.)
influences a model having a lesser probability, but wherein a model
having a lesser probability (.mu.) never influences a model having
a higher probability.
Inventors: |
Kronhamn; Thomas; (Halso,
SE) |
Correspondence
Address: |
ERICSSON INC.
6300 LEGACY DRIVE, M/S EVR 1-C-11
PLANO
TX
75024
US
|
Family ID: |
35929801 |
Appl. No.: |
11/995808 |
Filed: |
July 14, 2005 |
PCT Filed: |
July 14, 2005 |
PCT NO: |
PCT/IB2005/052341 |
371 Date: |
January 15, 2008 |
Current U.S.
Class: |
703/2 |
Current CPC
Class: |
G01S 13/723 20130101;
G06F 17/18 20130101 |
Class at
Publication: |
703/2 |
International
Class: |
G06F 17/10 20060101
G06F017/10 |
Claims
1.-15. (canceled)
16. A method of performing a sequence of measurements (z, R; M;
(t.sub.1,t.sub.2)t1, t2)) of at least one parameter (Pos; Vel; x,
P) and recursively performing predictions of at least the same or
at least another parameter (Pos, Vel; x, P), the prediction method
being based on, for a number of prediction periods, for instance,
corresponding to each possible measurement instance
(t.sub.k,t.sub.k+T.sub.p), defining a model set (PDF) having at
least two alternative models (PDF) having respective different mean
values (x.sub.i.sup.p, . . . ), respective covariance matrices
(P.sub.i.sup.p, . . . ) and corresponding respective probabilities
(.mu..sub.i.sup.p, . . . ), the models (PDF) approximating possible
outcomes, for instance, corresponding to various maneuvers in a
two-dimensional plane, the model set (PDF) having at least one
complementary (C) model and at least one sub (S) model, the method
having the steps of: based on at least on a first measurement
instance (M(t.sub.k); (k)), predicting the outcome (x, P) for at
least two models (C, S); after a subsequent measurement instance
(M(t.sub.k+T.sub.p) (k+Tp)) updating the models (C, S) for the
corresponding point in time, whereby the prediction made on the
basis of the first measurement instance is updated in the light of
the subsequent measurement instance; re-arranging at least one
model (C, S) for the subsequent measurement instance
(t.sub.k+T.sub.p) (k+Tp), whereby one updated model influences
another updated model, and wherein the step of re-arranging the S
model never influences the C model.
17. The method according to claim 16, wherein the model influence
involves that a first model j having a probability (.mu..sub.j)
changes the probability (.mu..sub.i) of a second model i according
to: .mu..sub.i.sup.a=.mu..sub.i.sup.u+.DELTA..mu..sub.ij,
.mu..sub.j.sup.a=.mu..sub.j.sup.u-.DELTA..mu..sub.ij where
.DELTA..mu..sub.ij=.kappa.(.mu..sub.j.sup.u-.mu..sub.i.sup.u) given
that .mu..sub.j.sup.u>.mu..sub.i.sup.u, wherein .kappa. is a
constant.
18. The method according to claim 17, wherein the proportionality
constants, .kappa..sub.C and .kappa..sub.S, are in the intervals
.kappa..sub.C.epsilon.(0,01;0,1) and
.kappa..sub.S.epsilon.(0,1;0,5),
19. The method according to claim 17, wherein for a one-dimensional
model set, the proportionality constant as
.kappa.=(.kappa..sub.C).sup.|i-j| where i-j is a natural number,
which corresponds to the model distance.
20. The method according to claim 17, wherein the step of
rearrangement is a one step procedure.
21. The method according to claim 16, wherein the model influence
involves that the state estimate (X) and the covariance (P) for a
given model is influenced by at least another model, whereby a
probability diffusion matrix applies, and
.mu..sup.a=M.sub.d.mu..sup.u whereby the state estimate and the
covariance is influenced according to the following relation x i a
= j m d , ij .mu. j u x j u / .mu. i a ##EQU00009## P i a = j m d ,
ij .mu. j u ( .DELTA. x ij .DELTA. x ij T + P j u ) / .mu. i a
##EQU00009.2## where .DELTA.x.sub.ij=x.sub.j.sup.u-x.sub.i.sup.a,
whereby the re-arranged state estimates, are influenced from all
models j with probabilities greater than model i.
22. The method according to claim 21, wherein the proportionality
constants, .kappa..sub.C and .kappa..sub.S, are in the intervals
.kappa..sub.C.epsilon.(0,01;0,1) and
.kappa..sub.S.epsilon.(0,1;0,5).
23. The method according to claim 21, wherein for a one-dimensional
model set, the proportionality constant as
.kappa.=(.kappa..sub.C).sup.|i-j| where i-j is a natural number,
which corresponds to the model distance.
24. The method according to claim 21, wherein the step of
rearrangement is a one step procedure.
25. The method according to claim 16, in which multiple predictions
are made for each measurement update, wherein only one
re-arrangement is necessary.
26. The method according to claim 16, whereby if the measurement
falls within the subset model (S), and the updated probability of
the subset model becomes highest, the respective models (C, S) do
not influence one another under the step of re-arrangement.
27. The method according to claim 16, whereby if the measurement
falls outside the subset model (S) and the updated probability of
the subset model becomes lowest, only the complementary model (C)
influences the subset (S) model under the step of
re-arrangement.
28. A method of performing a sequence of measurements (z, R; M;
(t.sub.1,t.sub.2)t1, t2)) of at least one parameter (Pos; Vel; x,
P) and recursively performing predictions of at least the same or
at least another parameter (Pos, Vel; x, P), the prediction method
being based on, for a number of prediction periods for instance
corresponding to each possible measurement instance
(t.sub.k,t.sub.k+T.sub.p), defining a model set (PDF) comprising at
least two alternative models (PDF) having respective different mean
values (x.sub.i.sup.p, . . . ), respective covariance matrices
(P.sub.i.sup.p, . . . ) and corresponding respective probabilities
(.mu..sub.i.sup.p, . . . ), the models (PDF) approximating possible
outcomes, for instance corresponding to various maneuvers in a
two-dimensional plane, the model set having exclusively
complementary (L, N, R) models, the method comprising the steps of:
based on at least on a first measurement instance (M(t.sub.k);
(k)), predicting the outcome (x, P) for at least two models (C, S);
after a subsequent measurement instance (M(t.sub.k+T.sub.p) (k+Tp))
updating the models (C, S) for the corresponding point in time,
whereby the prediction made on the basis of the first measurement
instance is updated in the light of the subsequent measurement
instance; re-arranging at least one model (C, S) for the subsequent
measurement instance (t.sub.k+T.sub.p) (k+Tp), whereby one updated
model influences another updated model, wherein for a given pair of
models within the model set (L, N, R), a model having a higher
probability (.mu.) influences a model having a lesser probability,
and wherein the step of re-arranging, for a given pair of models
within the model set (L, N, R), a model having a lesser probability
(.mu.) never influences a model having a higher probability.
29. The method according to claim 28, wherein the model influence
involves that a first model j having a probability (.mu..sub.j)
changes the probability (.mu..sub.i) of a second model i according
to: .mu..sub.i.sup.a=.mu..sub.i.sup.u+.DELTA..mu..sub.ij,
.mu..sub.j.sup.a=.mu..sub.j.sup.u-.DELTA..mu..sub.ij where
.DELTA..mu..sub.ij=.kappa.(.mu..sub.j.sup.u-.mu..sub.i.sup.u) given
that .mu..sub.j.sup.u>.mu..sub.i.sup.u, wherein .kappa. is a
constant.
30. The method according to claim 29, wherein the proportionality
constants, .kappa..sub.C and .kappa..sub.S, are in the intervals
.kappa..sub.C.epsilon.(0,01;0,1) and
.kappa..sub.S.epsilon.(0,1;0,5),
31. The method according to claim 29, wherein for a one-dimensional
model set, the proportionality constant as
.kappa.=(.kappa..sub.C).sup.|i-j| where i-j is a natural number,
which corresponds to the model distance (32).
32. The method according to claim 29, wherein the step of
rearrangement is a one step procedure.
33. The method according to claim 28, wherein the model influence
involves that the state estimate (X) and the covariance (P) for a
given model is influenced by at least another model, whereby a
probability diffusion matrix applies, and
.mu..sub.a=M.sub.d.mu..sup.u whereby the state estimate and the
covariance is influenced according to the following relation x i a
= j m d , ij .mu. j u x j u / .mu. i a ##EQU00010## P i a = j m d ,
ij .mu. j u ( .DELTA. x ij .DELTA. x ij T + P j u ) / .mu. i a
##EQU00010.2## where .DELTA.x.sub.ij=x.sub.j.sup.a-x.sub.i.sup.a,
whereby the re-arranged state estimates, are influenced from all
models j with probabilities greater than model i.
34. The method according to claim 33, wherein the proportionality
constants, .kappa..sub.C and .kappa..sub.S, are in the intervals
.kappa..sub.C.epsilon.(0,01;0,1) and
.kappa..sub.S.epsilon.(0,1;0,5),
35. The method according to claim 33, wherein for a one-dimensional
model set, the proportionality constant as
.kappa.=(.kappa..sub.C).sup.|i-j| where i-j is a natural number,
which corresponds to the model distance.
36. The method according to claim 33, wherein the step of
rearrangement is a one step procedure.
37. The method according to claim 28, in which multiple predictions
are made for each measurement update, wherein only one
re-arrangement is necessary.
38. The method according to claim 28, whereby if the measurement
falls within a given model (L, N, R), and the updated probability
of the model within the measurement becomes highest, the model for
which the measurement fell, influences the remaining models under
the step of re-arrangement.
39. The method according to claim 38, wherein models of higher
probabilities influence models of lesser probabilities.
40. The method according to claim 28, whereby if the measurement
falls between two given models (L, N, R), whereby the updated
probability of the models between the measurement fell becomes
equal, only the particular two models between which the
measurements fell, influences the remaining model or models under
the step of re-arrangement.
41. The method according to claim 16, being used for a radar
application (R), whereby at least the position (Pos) of an object
is sensed or derived in two dimensional coordinates and whereby
some of the probability models correspond to the associated
forthcoming position (Pos) in two dimensional coordinates.
42. The method according to claim 28, being used for a radar
application (R), whereby at least the position (Pos) of an object
is sensed or derived in two dimensional coordinates and whereby
some of the probability models correspond to the associated
forthcoming position (Pos) in two dimensional coordinates.
43. The method according to claim 33, wherein an element in the
diffusion matrix is limited according to
m.sub.dC,ij=min((.kappa..sub.C).sup.|i-j|max(.mu..sub.C,j.sup.u-.mu..sub.-
C,i.sup.u,0),.mu..sub.C,i.sup.u)/.mu..sub.C,j.sup.u.
Description
FIELD OF THE INVENTION
[0001] The present invention relates to prediction based on
stochastic models, and more particularly to recursive methods, that
is, prediction models repeatedly involving choosing among multiple
statistical prediction models based on measurement data.
[0002] More specifically, the present application may be applied in
a radar system predicting two-dimensional movements of objects as
seen from a birds view perspective, whereby the radar system is
measuring for instance the position of an object and is predicting
future values of the position and the velocity of the object.
BACKGROUND OF THE INVENTION
[0003] FIG. 1, is a birds view representation of a radar system R
observing the position of a flying object. A number of observations
are made by the radar system. As indicated in FIG. 1, the measured
position M may be associated with a given measurement accuracy,
which again may be approximated by a circular Gaussian probability
density function (PDF), two examples of which have been illustrated
in FIG. 5. The radius of the circular area corresponds to the
standard deviation of the Gauss distribution.
[0004] Having more measurements of the position at hand, a
probability density function Pos of the position and a probability
density function Vel for the estimated velocity of the object can
be predicted, please c.f. FIG. 1, whereby the estimated velocity
vector is used as a basis for the forthcoming prediction of the
position and velocity vector.
[0005] The present application is dealing with rendering such
predictions more accurate and reliable as well as less
computationally demanding.
[0006] In the following, a brief summary of the workings of the
known prediction methods vis-a-vis the prediction method of the
current invention will be made by referring to the exemplary
scenario of a radar system observing a flying object. In such a
system, the possible flight path is dependent on the capabilities
of the airplane and the "behavior" of the pilot, as defined by his
given task.
[0007] It should be understood, that the methods of the prior art,
as well as the methods of the invention, are not limited to any
particular application, such as to radar system applications. It
should moreover be noted that in the context of prediction models,
that the choice of measurement parameter is arbitrary. It should
also be noted that prediction models could be based on a
combination of sensed parameters. In the following description,
focus has been put on a system only sensing the position of an
object.
[0008] It should be noticed that a typical pulse-Doppler radar
system is capable of sensing the radial position of an object with
high accuracy. Therefore, the PDF may for a typical radar
application more accurately be approximated by elliptical PDF's as
indicated in FIG. 4. However, as can be readily understood,
predictions based on elliptical Gaussian probability functions are
complex computational tasks. Elliptical distributions are widely
used, but for illustrative simplicity circular PDF's are used in
this document.
[0009] In the following figures, the measured position will be
represented by an X and the measurement accuracy of the position
will be approximated by a Gaussian circular PDF. It should be
understood that in the following figures of the current document,
whenever a PDF is referring to circular representation, the radius
in question is proportional to the standard deviation of the PDF in
question.
Prediction Models--Model Sets
[0010] Modeling non-linear and non-Gaussian dynamic systems by
multiple models (MM) is a common procedure in Bayesian estimation
today, especially in high performance systems. In general, however,
methods range from the single model Extended Kalman Filter to
Particle filters, [see ref. 1]
[0011] Multiple dynamic models combined with a Markov mode switch
model is a particularly useful approach when the real world system
can be described as obeying the dynamics of some of the models for
a period of time and then switch to another model.
[0012] Typical applications for this type of modeling are tracking
of moving objects by sensor measurements, control of industrial
processes, etc.
[0013] In FIG. 6, a maneuver model based on a single Gaussian
circular PDF is chosen so as to represent the behavior of a flying
object, that is, for a given estimate of the current position and
velocity of a flying object the predicted position Cp and the
predicted velocity Cv are approximated. This model assumes that
acceleration; deceleration, left turn and right turn maneuvers are
equally possible, within a given prediction time.
[0014] In FIG. 8, an alternative model for the flying object
constructed from three Gaussian PDF's has been illustrated. It may
be assumed that the straight-ahead option N is more likely than the
left turn L and right turn R possibilities. Such a model may be
closer to reality when describing the flight path of for instance a
passenger traffic airplane. Suffix p and v corresponds to the
position and velocity.
[0015] As an alternative to the maneuver model discussed above
under FIG. 6, a multi hypothesis modeling formed by FIGS. 6 and 7
in combination comprising an outer complementary (C) model
enclosing an inner subset (S) model may be chosen.
[0016] The purpose of the totality of the complementary models
(C-model) is to represent all possible future actions by the flying
object. The likelihoods of the future predicted positions are, for
example, given by the PDF, represented by the circle Cp of FIG. 6.
Hence, the purpose of the exclusive complementary model set is to
cover an area of the two dimensional field by different models
overlapping one another as little as possible.
[0017] A subset model (S-model) is a model that represents a subset
of the system states as described by at least one C-model. The
purpose of the S-model is to represent a temporally dominant system
state(s). Often there is only one S-model. In FIG. 5, the Gaussian
PDF's of Sp and Cp have been illustrated. Each of the areas
represented by the Sp and Cp amounts to 1, if they are looked upon
as separate models. If they together are representing a
non-Gaussian PDF, they are given weights (probabilities) so that
the combined area amounts to 1. (The purpose of a S-model is to
cover the same field by overlapping models)
[0018] In the present context of a radar system observing the
two-dimensional flight path of a flying object, the complementary
model of FIG. 6 represents a maneuver model, while the sub-set
model of FIG. 7 represents a non-maneuver model.
[0019] For airplane applications, the FIGS. 6 and 7 models may be
weighted with probabilities 0.7 for the S-model and 0.3 (not shown)
for the C-model. This joint distribution would describe a pattern
in which a non-maneuver is more likely than a maneuver, which would
be the case for most airplane traffic behavior. It is noted, that
probabilities depend both on a priori assumptions and on how
accurate the modeling fits with the behavior as given by the
measurements. This may be calculated by the well known prior art of
Bayes' rule.
Recursive Modeling
[0020] Let us return to the prediction model based on three
Gaussian PDF's, c.f. FIG. 8, which have a total probability of 1.
Such a model has been implemented over consecutive measurement
occasions depicted in FIG. 9, whereby each Gaussian PDF, left,
right and straight ahead, by way of example, may be considered
equally likely.
[0021] A recursive method involves that the actual position and
velocity in two-dimensional space of the forthcoming PDF is
repeatedly determined by the last position of the measured object.
Consequently, the exemplary paths of FIG. 9 can depict consecutive
predictions.
[0022] When the measurement accuracy is good in relation to the
prediction model, the Gaussian PDF, which fits best on the recent
measurement, is chosen as a starting point for the next
prediction.
[0023] However, say a measurement falls in a left turn prediction
model; the recent measurement may not necessarily correspond with
reality. That is, from an a-posteriori point of view, the
assumption that the object actually took a left turn may be
rejected or confirmed or evaluated as to its likelihood by still
later measurements. This evaluation may be further evaluated by
still later measurement periods.
[0024] For this reason, an optimal recursive modeling may take into
account all previous maneuver possibilities. Such a recursive
modeling may be particular appropriate, if measurements are not
provided at every measurement instance or if measurements are
ambiguous due to the detection of other phenomena.
[0025] The idea of the known optimal MM (multiple model) tracking
method is to represent all possible mode combinations from the
start up to the present time and to calculate the probability of
each such combination by Bayes' rule. This leads to an
exponentially increasing number of filters (one for each
combination), as described by the "structure", [ref. 2]
(N.sub.e,N.sub.f)=(N.sub.m.sup.k-1,N.sub.m.sup.k)
where k is the number of measurement occasions, N.sub.m is the
number of models, N.sub.e is the number of estimates at the
beginning of each cycle and N.sub.f is the number of filters in the
algorithm.
[0026] It is readily understood, that it will become computational
burdensome to make evaluations based on more model update periods
by taking all possibilities into account. Moreover, some
possibilities will appear highly unlikely as measurements are
performed. Therefore, branches relating to older highly unlikely
branches may be abandoned from consideration--or pruned in order to
keep the number of filters within the limits of computational
possibilities. This means that combinations with low probabilities
are deleted (pruned) and combinations with almost equal state
estimates are combined (merged). This kind of control has little
influence on the joint PDF (probability density function) if the
probability threshold is low (and the probabilities are
re-normalized) and the merge method at least preserves the first
and second moments. In this way, however, there is no firm control
of the number of filters.
[0027] An effort to systemize the MHT concept, with a firm control
of the number of filters, is the Generalized Pseudo Bayesian
Estimators of order n (GPBEn) [see ref. 2]. These keep mode
combinations n time steps (measurement cycles) back. They have a
structure of
(N.sub.e,N.sub.f)=(N.sub.m.sup.n-1,N.sub.m.sup.n)
The number of filters is fixed.
[0028] The corresponding limitation on filters has been illustrated
in FIG. 10 for n=1, i.e. GPBE1.
[0029] The Interacting Multiple Model (IMM) algorithm has been
considered, for many applications, as constituting the best
compromise between performance and computational requirements. The
reason is that the IMM model interaction technique reduces the
computational structure to
(N.sub.e,N.sub.f)=(N.sub.m,N.sub.m)
[0030] In IMM, the models are "interacting" at each time step
before the prediction phase. This is reducing the number of
prediction filters to the same number as the number of models. The
interaction can be seen as a merge process governed by the Markov
mode switch model.
[0031] In general, the recursive, non-linear and non-Gaussian
estimation problem is a matter of representing a non-Gaussian and
non-stationary probability density function. The Multiple Model
(MM) approach, of different versions, is doing this as a sum of
Gaussian PDF's weighted according to their probabilities. After
each measurement, most methods--except the true optimal MM--have to
re-arrange (adapt) the models to better represent the PDF during
the prediction phase. This is because a limited number of filters
shall replace the optimal, exponentially increasing number of
filters.
[0032] We shall now look at the measurement, update, re-arrangement
and prediction stages of a sub-optimal MM-method such as the IMM
model, as exemplified by FIGS. 11, 12, 14 and 15, which is based on
the maneuver model of FIG. 8. In this exemplary scenario, the
measurement M is assumed to fall in the centre of the left turn
model.
[0033] In FIG. 11, after the measurement is completed at time t1,
the probability assessment for the maneuver at time t1 may be
re-evaluated such that the PDF model or models for which the
measurement likely occurred are considered more likely than assumed
a priori. Such a re-evaluation or update has been indicated in FIG.
12, where the left turn is assigned the highest probability, .mu.1,
the straight a head option a lower possibility .mu.2 and the right
turn is assigned a still lower possibility .mu.3, whereby the sum
of all possibilities remains 1. Moreover, according to the update
step, each respective model is displaced in two-dimensional space
in dependency of the measurement outcome. In the present example
the left turn model L1 is not displaced, as the measurement
happened to be in the centre of the left turn model. However, N1
and R1 are displaced in proportionality with the difference between
the measurement and the previous respective prediction model. A
darker hatching of the PDF corresponds to a higher probability
(this also applies to FIGS. 11-15 and 22-37).
[0034] FIG. 14, which follows the steps shown in FIG. 11 and FIG.
12 (of time t1), relates to the rearrangement according to
suboptimal MM-methods, such as the IMM method and the proposed
method. We shall now explain the mechanisms by studying, what would
happen if further predictions were made under the condition that no
more than one model may be used for prediction, per already used
model in the update phase (and not three models as shown
previously, FIG. 9), in order to keep down the computational
complexity, and if no re-arrangement process is applied. Such a
scenario has been indicated in FIG. 13, where models L2, M2 and R2
to time t2 are predicted having likelihoods corresponding to the
updated predictions at time t1 for L1, l2 and L3. It is seen that
the predictions for time t2, may spread in all directions.
[0035] The IMM method avoids the above situation by re-arranging
the already used models for time t1, after the measurement, by
displacing the centers of the Gaussian PDF's in two-dimensional
space as indicated from FIG. 12 and FIG. 14. All models are
displaced according to an influence from the other individual
models for the same time instance, whereby the model having the
greatest likelihood influences movement of a less likely model a
given amount and influences the movement of the least likely model
the most. This effect has been visualized in FIG. 14 showing the
displacements of L1 to L1', M1-M1' and R1-R1'. Moreover, the
probabilities (not shown) of the respective models are changed.
[0036] The result of the update, FIG. 12, and rearrangement, FIG.
14, has been shown in FIG. 15, where the placements of the models
are more narrowly centered around the recent measurement
direction.
[0037] In the above explanation, reference was made to a situation
where, the measurement clearly fell within the standard deviation
of one of the Gaussian models. This may of course not always be the
case, as shall be discussed in more detail later.
[0038] Further details relating to the processes of starting from
time k, predicting to and measuring at time k+Tp and updating the
models for time k+Tp for respectively the three maneuver model of
FIG. 8 have been shown in FIGS. 16 and 17. The process for updating
the model for the combined complementary/sub-model and of FIGS. 6
and 7 has been shown in more detail in FIGS. 18-21. In FIGS. 16-21
the indices are p--position, v--velocity; M--measurement, L--left;
N--straight on; R--right; C--complementary, S--subset;
u--update.
[0039] In FIG. 17, the relative position of PDF Lp has been
indicated together with Lpu in order to demonstrate the model
update. The same applies for FIGS. 19 and 21, respectively with
regard to FIGS. 18 and 20, respectively.
General Model--State Vector
[0040] In the foregoing, focus has been on a two-dimensional radar
application. However, as mentioned above, the method according to
the invention is also applicable to other systems. Therefore, the
method shall be described with regard to a general state vector,
which description is well known in the art.
[0041] The dynamic system is described at the time instant k by the
state vector x(k).
[0042] A moving object may have the state vector elements
x ( k ) = [ x y x . y . ] ( 1 ) ##EQU00001##
in a horizontal, 2-dimensional Cartesian coordinate frame.
[0043] The state of the dynamic system may be estimated by
measurements obtained from a sensor (e.g. radar).
[0044] The estimated state vector may have the same form as above,
or [0045] {circumflex over (x)}(k|k), where the "hat" separates the
estimate from the true value, the first time suffix shows the time
of the estimate and the second time suffix shows the last
measurement used. In this way, [0046] {circumflex over (x)}(k+1|k)
is an estimate of the state vector at time k+1, where measurements
up to and including time k have been used. This last expression is
also called the (one step) predicted value from time k. For
simplicity the hat may be omitted in many cases.
[0047] State vector estimates may also be described by an
uncertainty measure, describing the accuracy of the estimate. In
the usual case of Gaussian uncertainties (probability density
functions) they are completely described by the covariance matrix
P. For the state estimate above the covariance matrix is
P ( k k ) = [ p xx p xy p x x . p x y . p yx p yy p y x . p y y . p
x . x p x . y p x . x . p x . y . p y . x p y . y p y . x . p y . y
. ] ( 2 ) ##EQU00002##
where the diagonal elements are the variances in the respective
coordinate, i.e.
6.sub.ii=.sigma..sub.ii.sup.2, (3)
where i is the row and column number, and the off-diagonal elements
are the covariances of the respective state elements of that row
and column, respectively, i.e.
p.sub.ij=.rho..sub.ij.sigma..sub.i.sigma..sub.j, (4)
where .rho. is the correlation coefficient, or the standard
deviation and i and j are the row and column numbers,
respectively.
[0048] The upper left 2.times.2 matrix, of P(k|k), represents the
2-dimensional position uncertainty and can be visualized as an
ellipse in the xy-plane, centered at the position estimate. In the
same way, the lower right 2.times.2 matrix is the 2-dimensional
velocity uncertainty. This may be visualized as an ellipse "on top
of the velocity vector". This has been illustrated in FIG. 38.
Block Diagram of Recursive Filter
[0049] FIG. 3 shows a block diagram of a general, recursive filter.
The figure shows the signal flow and the functional blocks for one
cycle of calculations. The cycle duration may be any time T,
constant or changing.
[0050] In FIG. 3, the measurement and state vectors are denoted z
and x, with co-variance matrices R and P, respectively. The filter
comprises two mandatory blocks, state and co-variance update block
"State & cov. update", (7) and (8)--referring to equations
given in the following--and state and covariance prediction block,
"State & cov. pred.", (13), and an optional block expected
measurement and gating block, "Exp. Meas., Gating". The latter
block is in some applications used for fencing off unwanted
measurements. It calculates the expected value of the measurement,
(6), and an area/volume within which the actual measurement should
fall.
[0051] The basic function of the recursive filter is as follows: At
time index k, a measurement z(k), R(k) (5) occurs. A previously
predicted estimate, i.e. predicted before time index k for time
index k, denoted x(k|k-1), P(k|k-1) is provided to the expected
measurement and gating block together with the corresponding
measurement for time index k, z(k), R(k). The result is provided to
the state and covariance update block to which a further input of
the measurement z(k), R(k) is made. An updated estimate for x(k|k),
P(k|k) (7) is provided. Finally, the updated result is processed in
the state and covariance pre-diction block, for predicting the
state variable and covariance for x(k|k+1), P(k|k+1) (13) for the
successive time index. The process is iterated with the result and
measurement for the next time index.
The IMM Algorithms
Update:
[0052] Given a measurement (vector) z(k), at time k, which relates
to the true state vector x(k) as given by the (linear) equation
z(k)=Hx(k)+v(k) (5)
where [0053] H is the so-called measurement matrix, [0054] v is the
measurement noise.
[0055] The accuracy of the measurement is in this formulation
described by the measurement noise covariance matrix R.
[0056] The expected measurement is the part of the predicted state
vector that is measured, e.g. the position in this case,
z(k|k-1)=Hx(k|k-1) (6)
[0057] The update equations, for a model i, may be written as
x.sub.i(k|k)=x.sub.i(k|k-1)+K.sub.i(k)(z(k)-H.sub.ix.sub.i(k|k-1))
P.sub.i(k|k)=(I-K.sub.i(k)H.sub.i)P.sub.i(k|k-1)(I-K.sub.i(k)H.sub.i)'+K-
.sub.i(k)RK.sub.i(k)' (4)
[0058] The update gain matrix K.sub.i may be calculated as the
Kalman gain
K.sub.i(k)=P.sub.i(k|k-1)H.sub.i'[H.sub.iP.sub.i(k|k-1)H.sub.i'+R].sup.--
1 (8)
[0059] The prediction and update equations given above constitute
the well-known Kalman filter equations. (Especially the covariance
update equation above can be written in a number of different, but
equivalent, forms).
[0060] In non-linear applications, e.g. with the left and right
turn models, these equations may be slightly modified from the
original Kalman filter equations for linear models, to the equally
well known form of the so called Extended Kalman filter
equations.
[0061] The update of the model probabilities follows the well-known
Bayes' rule
.mu. i ( k k ) = li ( z ( k ) i ) .mu. i ( k k - 1 ) / i li ( z ( k
i ) .mu. i ( k k - 1 ) ( 9 ) ##EQU00003##
where li(z(k|i) means the likelihood of the measurement z given the
model i.
Probability Prediction
[0062] The probability prediction may be written as
.mu.(k+1|k)=M.sub.p.mu.(k|k) (10)
where M.sub.p is the probability prediction matrix, in most cases
based on a Markov mode switch model.
Model Interaction
[0063] The model data interaction is governed by the probability
prediction.
[0064] For each model i:
x i I ( k ) = j N m ij .mu. j ( k k ) x j ( k k ) / .mu. i ( k + 1
k ) P i I ( k ) = j N m ij .mu. j ( k k ) ( .DELTA. x ij .DELTA. x
ij ' + P j ( k k ) ) / .mu. i ( k + 1 k ) ( 11 ) ##EQU00004##
where
.DELTA.x.sub.ij=x.sub.j(k|k)-x.sub.i.sup.I(k) (12)
and N is the number of models and m.sub.ij are the elements of the
probability prediction matrix M.sub.p.
Model Data Prediction
[0065] The standard, time discrete, state vector prediction, for a
linear model, from time instant k to time instant k+1 may be
written as
x.sub.i(k+1|k)=.PHI..sub.ix.sub.i(k|k)
P.sub.i(k+1|k)=.PHI..sub.iP.sub.i(k|k).PHI..sub.i'Q.sub.i (13)
where [0066] i is the model number [0067] .PHI. is the
deterministic prediction model, e.g. straight line prediction,
[0068] .PHI.' is the transpose of .PHI., [0069] Q is the stochastic
prediction model, often called the process noise co-variance
matrix, used to model object maneuvers. For a model of a
non-maneuvering object, this is a null-matrix.
[0070] For a non-linear model, such as left or right turn models,
these equations take slightly different forms. This is well known
from the theory of the so-called Extended Kalman filter.
Block Diagram of the IMM Filter
[0071] FIG. 39 shows a block diagram of the IMM filter for a
two-model case. FIG. 39 shows the signal flow and the functional
blocks for two cycles of calculations. Time is denoted by the time
index k, when the measurements occur. The cycle duration may be any
time Tp, constant or changing. In FIG. 39, the measurement and
state vectors are denoted z and x, with co-variance matrices R and
P, respectively. The indices 1 and 2 refer to the two different
models.
[0072] FIG. 39 shows the model probabilities .mu. and the
functional blocks for the probability update and prediction, (9)
and (10). FIG. 39 also shows the functional block for the "State
& covariance interaction", (11) and that this interaction is
"governed" by the probability prediction. This is the fundamental
process of the IMM filter.
Problems with Existing Solutions
[0073] The main problem with high performance, sub-optimal Bayesian
solutions (e.g. non-optimal MM, GPBEn for large n) is the
computational burden. IMM is considered the best "cost effective"
(i.e. performance vs. computations) solution today for many
applications.
[0074] The mode merging coupled to the mode switch model of the IMM
has many advantages, but the method is also associated with
disadvantages. For instance the IMM comes from that the mode
merging is coupled with the mode switch model. The IMM does not
cause problems when prediction is done just once per measurement
update cycle. However, there are many applications where this is
not the case. Such applications may be: [0075] Sensors with
probability of detection <1 [0076] Uncertain association; when
multiple predictions have to be performed to find the appropriate
measurement data. [0077] Adaptive sensors, where repeated
predictions are made to find out the best measurement time. [0078]
Multi-sensor applications, where the expected measurement time is
unknown and multiple predictions have to be done to find the
appropriate set of measurement data [0079] Data prediction, in
general, that is not measurement oriented.
[0080] Another problem with IMM is that the mutual interaction
between a "maneuver model" and a "non maneuver model". When the
system state is "non maneuver", makes the pre-diction volume
smaller than anticipated from the design of the maneuver model.
Should a maneuver occur, this may lead to track loss, if not the
maneuver gate has been enlarged by other means [see ref. 3, Section
4.5.6, page 232)].
REFERENCES
[0081] [1] B. Ristic, S. Arulampalam, and N. Gordon, Beyond the
Kalman Filter, Particle Filters for Tracking Applications. Artech
House, 2004. [0082] [2] Y. Bar-Shalom, X. R. Li, and T.
Kirubarajan, Estimation with Applications to Tracking and
Navigation. New York: John Wiley & Sons, 2001. [0083] [3] S.
Blackman, and R. Popoli, Design and Analysis of Modern Tracking
Systems, Artech House, 1999. [0084] [4] H. A. P. Blom, "An
Efficient Filter for Abruptly Changing Systems," in Proc. 23rd IEEE
Conf. Decision and Control, Las Vegas, Nev., December 1984.
SUMMARY OF THE INVENTION
[0085] It is a first object of the invention to provide a
prediction method based on a model set comprising at least one
complementary model and at least one sub-model, which method may
perform more accurately predictions for many applications and which
is less computational burdensome than known methods in many
applications.
[0086] This object has been achieved by the subject matter defined
in claim 1.
[0087] It is a second object of the invention to provide a
prediction method based on a model set comprising exclusively
complementary models, which method may perform more accurately
predictions for many applications and which is less computational
burdensome than known methods in many applications.
[0088] This object has been achieved by the subject matter defined
in claim 2.
[0089] It is a further object of the invention to set forth a
method that further ameliorates the computational burden.
[0090] This object has been achieved by the subject matter defined
by claim 3.
[0091] Further advantages will appear from the following detailed
description of the invention.
BRIEF DESCRIPTION OF THE DRAWINGS
[0092] FIG. 1 shows measurement tolerances relating to a radar
system measuring the position of a flying object and corresponding
predictions of future position and velocity of the flying
object,
[0093] FIG. 3 shows the calculation process for a general recursive
model,
[0094] FIG. 4 relates to a radar system, in which measurements are
associated with elliptical Gaussian measurement tolerances,
[0095] FIG. 5 shows well-known exemplary Gauss distributions,
[0096] FIG. 6 shows a maneuver model,
[0097] FIG. 7 shows a non-maneuver model,
[0098] FIG. 8 shows a maneuver model comprising 3 complementary
models,
[0099] FIG. 9 shows a multiple model,
[0100] FIG. 10 shows model merging according to the GPBE1
model,
[0101] FIG. 11-15 show recursive phases involving prediction,
measurement, update, re-arrangement and prediction of models,
[0102] FIG. 16-17 show model prediction and update for the
three-maneuver model of FIG. 8
[0103] FIG. 18-21 show model prediction and update for a model
combination of the complementary model of FIG. 6 and the sub-set
model of FIG. 7,
[0104] FIG. 22-25 show model prediction, update and re-arrangement
for the invention vis-a-vis the known IMM model for combined C and
S model in which a measurement falls within the S model
prediction,
[0105] FIG. 26-29 show model prediction, update and re-arrangement
for the invention vis-a-vis the known IMM model for combined C and
S model in which a measurement falls within the C model
prediction,
[0106] FIG. 30-33 show model prediction, update and re-arrangement
for the invention vis-a-vis the known IMM model for the
three-maneuver model in which a measurement is clearly falling
within the left turn prediction model,
[0107] FIG. 34-37 show model prediction, update and re-arrangement
for the invention vis-a-vis the known IMM model for the
three-maneuver model in which a measurement is falling between the
left turn and straight on prediction models,
[0108] FIG. 38 shows the position and velocity vector e.g. relating
to a 2-dimensional depicting radar system,
[0109] FIG. 39 shows the calculation process for the known IMM
model for a first measurement instance for which a measurement is
made and for a subsequent measurement instance for which a
measurement is missed,
[0110] FIG. 40 shows the calculation process for the invention for
a first measurement instance for which a measurement is made and
for a subsequent measurement instance for which a measurement is
missed, and
[0111] FIG. 41 shows a further embodiment of the invention.
DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS OF THE INVENTION
[0112] The invention concerns a new concept--denoted the
"probability mass diffusion" (PMD) principle--for re-arranging the
model set after the measurement updates.
[0113] The present invention, shall initially be explained in more
detail by focusing on the computational workings and effects in
close relation to the IMM method as applied to a number of
exemplary radar applications, while later expanding the description
to general applications where recursive estimation of dynamic
systems/signals are used.
[0114] For this purpose we shall initially focus on four examples,
in which the same complementary model/subset model is used for the
PMD method and the IMM method, respectively and where the
measurements fall either clearly within a given model or in an
ambiguous area between models, respectively.
[0115] The indices are the same as defined above for FIGS. 16-21,
except that being expanded with suffix a denoting the re-arranged
model sets.
[0116] The IMM and the proposed method are equal in the measurement
and update phases. The differences in the following re-arrangement
(Interaction and adaptation respectively) will be described in the
following.
[0117] Among others, the proposed method and the IMM differ in the
manner the re-arrangement phase (Interaction and adaptation
respectively) is carried out and in the amounts of the influences,
which will be described in the following.
[0118] FIGS. 22-25 relate to the subset/complementary model for IMM
and PMD for the case that a measurement M falls within the subset
model Sp. FIGS. 22 and 23 relate both to PMD and IMM, whereas FIG.
24 relates to IMM only and FIG. 25 relates to PMD only.
[0119] In FIG. 22, the velocity and position of the object are
predicted according to a respective subset-model and a respective
complementary model. According to the example, the measurement of
the position M falls within both the complementary model and the
supplementary position models.
[0120] In FIG. 23, the position and velocity of both the
complementary models Cpu and Cvu and the supplementary models Spu
and Svu are displaced according to the position of the measurement.
The standard deviations of all models are significantly reduced in
relation to the predicted values. Since the measurement fell within
the predicted models, the areas of the updated models will both
contain the position of the measurement. The exact relations for
the model updates can be formulated according to standard Kalman
filter algorithms [2,3].
[0121] In FIG. 24, the model re-arrangement according to the known
IMM model is illustrated.
[0122] The respective complementary model, Cpu and Cvu, and the
respective supplementary model, Spu and Svu, both influence one
another, such that the values and the probabilities of Cpa and Cva
as well as Spa and Sva are changed.
[0123] According to the PMD method, no interaction of models takes
place in this given scenario, which means that Cpa and Cva are
equal to Cpu and Cvu, respectively, and that Spa and Sva are equal
to Spu and Svu, respectively. This is true for the case that the
probability of the subset model is higher than the probability of
the complementary model, which is the normal outcome when the
measurement was obtained within the areas of both models. However,
there may be marginal cases and sometimes also a dependence on
previous measurements that not lead to this situation.
[0124] Hence, if the measurement falls within the subset model, S,
and the updated probability of the subset model becomes highest,
the respective models, C, S, do not influence one another under the
step of re-arrangement.
[0125] The arrows in FIGS. 22-37 indicate which model or models
influence another or other models. Please note that the
displacement of the influenced model may move in another direction
than indicated by the arrow.
[0126] FIGS. 26-29 relate to the subset/complementary model for IMM
and PMD for the case that a measurement M falls outside the subset
model Sp, but within the complementary model. FIGS. 26 and 27
relate both to PMD and IMM, whereas FIG. 28 relates to IMM only and
FIG. 29 relates to PMD only.
[0127] In FIG. 27 it is seen that the complementary model is
substantially displaced, while the subset model is less affected.
The reason is that the design of the complementary model makes it
more willing for large displacements (as long as they are within
the area Cp). The subset model design makes it not willing for
displacements outside the area Sp. The update is performed
according to the relations mentioned above.
[0128] In FIG. 28, relating to the known IMM model it appears that
both models mutually influence one another.
[0129] In FIG. 29 though, only the complementary model influence
the subset-model.
[0130] Hence, if the measurement falls outside the subset model, S,
and the updated probability of the subset model becomes lowest,
only the complementary model, C, influences the subset, S, model
under the step of re-arrangement.
[0131] FIGS. 30-33 relate to a three maneuver Gaussian model for
IMM and PMD for the case that a measurement M falls within one of
the maneuver models. FIGS. 30 and 31 relate both to PMD and IMM,
whereas FIG. 32 relates to IMM only and FIG. 33 relates to PMD
only.
[0132] In FIG. 31, the model update, which is common both for the
IMM and the PMD methods are shown. It is seen that the models are
displaced in direction of the measurement as indicated, while the
probabilities are also updated so that the given maneuver model,
within which the measurement fell, gains the highest
probability.
[0133] In FIG. 32, relating to IMM, it is illustrated that all
models influence one another, both with respect to displacement of
models and with respect to re-evaluation of probabilities.
[0134] In FIG. 33, relating to PMD, it is illustrated that only the
models within which the measurement fell influence the remaining
models, both with respect to displacement of models and with
respect to re-evaluation of probabilities. This shall be understood
as the PMD-principle for C-models, that models with higher
probability influences models with lower probabilities.
[0135] In the FIG. 33 scenario, models of higher probabilities
influence models of lesser probabilities. If three models of
gradual probabilities occur, the model of highest probability
influences the two other models, while the model of the medium
probability influences the model of least probability.
[0136] Hence, if the measurement falls within a given model, L, N,
R, and the updated probability of the model within the measurement
becomes highest, only the model for which the measurement fell,
influences the remaining models under the step of re-arrangement.
As can be understood, the number of required computational
operations is decreased.
[0137] FIGS. 34-37 relate to a three maneuver Gaussian model for
IMM and PMD for the case that a measurement M falls between two
maneuver models. FIGS. 34 and 35 relate both to PMD and IMM,
whereas FIG. 36 relates to IMM only and FIG. 37 relates to PMD
only.
[0138] In FIG. 35, the model update, which is common both for the
IMM and the PMD methods are shown. It is seen that all models are
displaced in direction of the measurement M as indicated, while the
probabilities are also updated so that the two given maneuver
models, between which the measurement fell, gain the highest
probabilities.
[0139] In FIG. 36, relating to IMM, it is illustrated that all
models influence one another, both with respect to displacement of
models and with respect to re-evaluation of probabilities.
[0140] In FIG. 37, relating to PMD, it is illustrated that only the
models between which the measurement fell influence the remaining
model, both with respect to displacement of models and with respect
to re-evaluation of probabilities. This shall be understood as the
PMD-principle for C-models, that models with higher probability
influences models with lower probabilities and that in this case
two of the models have the same probability.
[0141] Hence, if the measurement falls between two given models, L,
N, R, whereby the updated probability of the models between the
measurement fell becomes equal, only the particular two models
between which the measurements fell, influences the remaining model
or models under the step of re-arrangement. As can be understood
the number of required computational operations are decreased.
Terminology
[0142] We shall now describe the inventive method in general terms
replacing the terms referring to the various PDF's of FIGS. 16-37
with general notions of mean value, standard deviation and
probability of corresponding PDF's.
[0143] The following explanation for a left turn maneuver PDF is
given, noting that a full explanation for all parameters would not
be necessary:
TABLE-US-00001 re-arrangement: adaptation (a)/ prediction update
interaction (I) Lpp, Lvp Lpu; Lvu Lpa; Lva PDF parameters
x.sub.i.sup.P x.sub.i.sup.u x.sub.i.sup.I, x.sub.i.sup.a mean
value/state vector P.sub.i.sup.P P.sub.i.sup.u P.sub.i.sup.I,
P.sub.i.sup.a covariance matrix .mu..sub.i.sup.P .mu..sub.i.sup.u
.mu..sub.i.sup.I, .mu..sub.i.sup.a probability
[0144] In the following, suffix I relates to the parameters used
under re-arrangement according to the IMM method, while suffix a
relates to the re-arrangement under the PMD method according to the
invention, and suffix i relates to any given model/PDF
(subset/complementary, exclusively complementary).
[0145] In the description of the prior art, the common notation of
time has been used, denoting the time instants of measurements (and
data updates) with the time index k. A constant time T between
these instants is often assumed. In the description of the
invention, the more general notation of a time instant as t.sub.k
and the time between measurements as T.sub.p is applied. Hence in
the present context, a time instant k corresponds to t.sub.k and
k+1 to t.sub.k+T.sub.p. The use of time indices k and k+1, in the
descriptions of the prior art, have the corresponding notation u
for k|k and p for k+1|k in the descriptions of the invention.
Model Adaptation by Probability Mass Diffusion; Basic Process
[0146] One aspect of the probability mass diffusion according to
the present invention is that probability mass is flowing from
models with higher probabilities to models with lower
probabilities. Given two models, where .mu..sub.2>.mu..sub.1
that is:
.mu..sub.1.sup.a=.mu..sub.1.sup.u+.DELTA..mu..sub.12
.mu..sub.2.sup.a=.mu..sub.2.sup.u-.DELTA..mu..sub.12 (14)
where index a indicates the re-arranged probabilities and index u
indicates the probabilities obtained at the model update by a
measurement.
[0147] The flow is a function of the difference in probability.
According to a preferred embodiment of the invention, the amount is
directly proportional, i.e. for the case above
.DELTA..mu..sub.12=.kappa.(.mu..sub.2.sup.u-.mu..sub.1.sup.u)
(15)
where .kappa. is the diffusion constant.
[0148] The influence on the model data is
x.sub.1.sup.a=(.mu..sub.1.sup.ux.sub.1.sup.u+.DELTA..mu..sub.12x.sub.2.s-
up.u')/.mu..sub.1.sup.a
x.sub.2.sup.a=x.sub.2.sup.u (16)
and
P.sub.1.sup.a=(.mu..sub.1.sup.u(.DELTA.x.sub.11.DELTA.x.sub.11'+P.sub.1.-
sup.u)+.DELTA..mu..sub.12(.DELTA.x.sub.12.DELTA.x.sub.12'+P.sub.2.sup.u))/-
.mu..sub.1.sup.a
P.sub.2.sup.a=P.sub.2.sup.u (17)
where .DELTA.x.sub.11=x.sub.1.sup.u-x.sub.1.sup.a
.DELTA.x.sub.12=x.sub.2.sup.u-x.sub.1.sup.a' (18)
[0149] Of importance for the estimation according to the invention
is also the view of models as either C- or S-models, as described
in the previous section. For C-models the diffusion may go in any
direction as given by the probability gradient. Between C- and
S-models there can only be a flow from the C- to the S-models, not
vice versa. The diffusion constant, .kappa..sub.C, is low for the
flow between C-models, as they should be as independent as
possible. The diffusion constant, .kappa..sub.S, is high for the
flow from C- to S-models, as the S-model always should be within
the bounds of the C-model(s). These constants are design parameters
and should be chosen to best suit the given application. Typical
values may be in the intervals
.kappa..sub.C.epsilon.(0.01,0.1) (19)
and
.kappa..sub.S.epsilon.(0.1,0.5). (20)
[0150] The re-arrangement process described above can, according to
a further embodiment, be expanded to work over a large set of
models (C-models), mimicking a diffusion process even in this
respect. Such a re-arrangements process for a large group of models
may be realized in different ways. One way is to start at the model
with the highest probability, let it "influence" its nearest
neighboring models and then let those models influence the next
closest of the models in the probability gradient direction, like a
"waterfall" (or rings in the water), until the entire set is
covered. Another preferred way, computationally, according to the
invention is a "one-step method", described below, where the
"influence" is rendered dependent on a parameter which we shall
denote as the "model distance".
[0151] In summary, the process can be described as:
[0152] A method of performing a sequence of measurements, z, R; M;
(t.sub.1,t.sub.2), of at least one parameter and recursively
performing predictions of at least the same or at least another
parameter, the prediction method being based on--for a number of
prediction periods for instance corresponding to each possible
measurement instance (t.sub.k,t.sub.k+T.sub.p)--defining a model
set (PDF) comprising at least two alternative models (PDF) having
respective different mean values (x.sub.i.sup.p, . . . ),
respective covariance matrices (P.sub.i.sup.P, . . . ) and
corresponding respective probabilities (.mu..sub.i.sup.p, . . . ),
the models (PDF) approximating possible outcomes, for instance
corresponding to various maneuvers in a two-dimensional plane.
[0153] The method comprising the steps of [0154] based on at least
on a first measurement instance (M(t.sub.k)), predicting the
outcome (x, P) for at least two models (C, S), [0155] after a
subsequent measurement instance (M (t.sub.k+T.sub.p)) updating the
models (C, S) for the corresponding point in time, whereby the
prediction made on the basis of the first measurement instance is
updated in the light of the subsequent measurement instance, and
[0156] re-arranging at least one model (C, S) for the subsequent
measurement instance (t.sub.k+T.sub.p), whereby one updated model
influences another updated model.
[0157] For a model set comprising at least one complementary (C)
model and at least one sub (S) model, under the step of
re-arranging the S model never influences the C model.
[0158] For a model set comprising exclusively complementary (L, N,
R) models, under the step of re-arranging, for a given pair of
models within the model set (L, N, R), a model having a higher
probability (.mu.) influences a model having a lesser probability,
but wherein a model having a lesser probability (.mu.) never
influences a model having a higher probability.
[0159] The "re-arrangement", or adaptation, of models according to
the methods above fulfils the following requirements of the
invention of altering the PDF as little as possible and of leaving
the models/filters independent of each other as much as possible.
The rational of these requirements can be understood from the
optimal MM where there is no re-arranging of filters as all mode
sequences are kept and from, sub-optimal solutions, where the
re-arranging is a "necessary evil" that secure that the correct
mode sequence is reasonably well represented.
[0160] For further aspects of the invention it applies that a first
model j having a probability (.mu..sub.j) changes the probability
(.mu..sub.i) of a second model i according to:
.mu..sub.i.sup.a=.mu..sub.i.sup.a+.DELTA..mu..sub.ij,
.mu..sub.j.sup.a=.mu..sub.j.sup.u-.DELTA..mu..sub.ij (21)
where
.DELTA..mu..sub.ij=.kappa.(.mu..sub.j.sup.u-.mu..sub.i.sup.u)
(22)
given that .mu..sub.j.sup.u>.mu..sub.i.sup.u, wherein .kappa. is
a constant.
Model Adaptation by Probability Mass Diffusion; Matrix Form
[0161] The step of model update and the step of model prediction
according to the invention take the same form as described for IMM
in the background of the invention.
[0162] In the following, the step of model adaptation according to
the invention that corresponds to the step of model interaction of
the IMM is described.
[0163] The equations describing the basic process may be given in
matrix form for any given number of models, as an alternative to
the previously indicated equations (14)-(18), whereby suffix
p=predicted, /=interacted, a=adapted and u=updated probabilities.
Matrix M.sub.p is the probability prediction matrix and M.sub.d is
the probability diffusion matrix.
[0164] The key is to write the probability mass diffusion in matrix
form. We introduce the probability diffusion matrix M.sub.d,
defined as
PA=Md/U, (23)
consequently, the complete cycle of probability calculations can be
written as
pP=M pPa=MpMd PU (24)
[0165] The corresponding expression for IMM is:
pp=pI=MPpu (25)
[0166] With this notation the PMD model adaptation may be written
in the same form as the IMM interaction, but now with the
probability diffusion matrix.
x i a = j m d , ij .mu. j u x j u / .mu. i a ( 26 ) P i a = j m d ,
ij .mu. j u ( .DELTA. x ij .DELTA. x ij T + P j u ) / .mu. i a ( 27
) ##EQU00005##
where
A=XuX. (28)
[0167] The above equations corresponds to the IMM-interaction
x i I = j m p , ij .mu. j u x j u / .mu. j I ( 29 ) P i I = j m p ,
ij .mu. j u ( .DELTA. x ij .DELTA. x ij T + P j u ) / .mu. i I , (
30 ) ##EQU00006##
where
Axu=XJU-xi. (31)
[0168] The detailed definition of the probability diffusion matrix
M.sub.d based on the one-step method given above is as follows:
[0169] A set of models, designed to represent a given PDF, is most
likely ordered along some main system parameter(s). In the case
used in the invention, this is the turn rate .omega. of the
prediction model. In the three-model case, the three C-models
having turn rates of (-.omega.,0,.omega.) are given the model
indices i.epsilon.(1,2,3). The "model distance" d.sub.m is defined
as
d.sub.m=|i-j| (32)
for any two (selected) models in the set. A similar distance
measure should always be possible to define even in
multi-dimensional sets of models.
[0170] According to the invention, the probability mass diffusion
is performed in one step and the diffusion constant for C-models is
defined as
.kappa.=(.kappa..sub.C).sup.|i-j|, (33)
where i and j are the model numbers defined above.
[0171] The diffusion equations (14)-(18) may be written in matrix
form for a general number of S- and C-models as described
below.
[0172] First, the diffusion matrix, M.sub.d, and the probability
vector, .mu., are subdivided into the parts related to the S- and
C-models respectively. That is
M d = [ I M dS 0 M dC ] and ( 34 ) .mu. = [ .mu. S .mu. C ] ( 35 )
##EQU00007##
[0173] The elements of M.sub.d are, for the case of a single
S-model:
I=1 (36)
and M.sub.dS is the row matrix
M.sub.dS={m.sub.dS,j}.sub.j=1.sup.N.sup.C, (37)
where N.sub.C is the number of C-models and where the elements are
of the form:
m.sub.dS,j=.kappa..sub.Smax(.mu..sub.C,j-.mu..sub.S,0)/.mu..sub.C,j.
(38)
[0174] A single S-model can be chosen to be influenced either by
one of the C-models, by each of the C-models individually or by the
joint C-model estimate. As described, a single S-model is
influenced by each of the C-models individually.
[0175] The elements of the matrix M.sub.dC are of the form
m.sub.dC,ij=(.kappa..sub.C).sup.|i-j|max(.mu..sub.C,j.sup.u-.mu..sub.C,i-
.sup.u,0)/.mu..sub.C,j.sup.u (39)
for i.noteq.j and where i and j are C-model numbers. The diagonal
elements of the sub-matrix M.sub.dC are
m dC , ii = 1 - ( m dS , i + k = 1 N C m dC , ki ) . ( 40 )
##EQU00008##
[0176] These equations show the basic diffusion process as
described above. However, a minor modification is shown in the
section below.
Limitation of Transferred Mass
[0177] When the transferred probability mass is far greater than
the probability of the recipient, the resulting effect on the state
estimate of the recipient model is large and in the limit it will
become equal to the estimate of the donor model estimate. If the
diffusion process is performed as a one-step procedure, this may
lead to that the order, or in other words, the spatial
relationships, of the models may be changed. Here is shown a
preferred way of executing a limitation of diffused probability
mass that will preserve the order of the models. The equation (39)
above is now written
m.sub.dC,ij=min((.kappa..sub.C).sup.|i-j|max(.mu..sub.C,j.sup.u-.mu..sub-
.C,i.sup.u,0),.mu..sub.C,i.sup.u)/.mu..sub.C,j.sup.u (41)
[0178] Various steps of calculating this equation have been shown
graphically in FIG. 41.
Block Diagram of the Invention
[0179] FIG. 40 shows a block diagram of the invention for a
two-model case closely corresponding to the IMM method depicted in
FIG. 39.
[0180] The signal flow graph, depicted within the functional blocks
Prob. mass diffusion and State & cov. adaptation, refers to
first scenario with one C-model and one S-model, where model 1
(.mu..sub.1) is the C-model and where the C-model has the highest
probability. The signal flow graph may also refer to a scenario
with two C-models where model 1 (.mu..sub.1) has the highest
probability.
[0181] Comparing the first measurement cycles of FIG. 39 and FIG.
40, it appears that the probability mass diffusion and the state
and covariance adaptation according to the invention are separated
from the probability prediction.
ADVANTAGES OF THE INVENTION
[0182] The main advantages of the invention are: [0183] the method
requires less computations (than IMM), in many applications, for
two reasons: [0184] The diffusion matrix always contains a number
of zero elements. This is not always the case for the corresponding
operation in IMM [0185] Separated model adaptation and model
prediction are separated saves computations when multiple
predictions are made per measurement occasion [0186] Avoidance of
too small track gates (small track gates described by S. Blackman,
and R. Popoli, Design and Analysis of Modern Tracking Systems,
Artech House, 1999). In other words, a better prediction of the
spatial probability density function is obtained. [0187] More
accurate results (than IMM) are obtained, as confirmed by extensive
simulations.
Computational Aspects
[0188] The second measurement cycle of FIG. 39 is an illustration
of the IMM recursive loop processing for a situation where at time
k+Tp, a measurement is missing, whereby two PDF's, suffix 1 and 2,
are used for modeling, e.g. a complementary/subset model
scenario.
[0189] The second measurement cycle of FIG. 40 is an illustration
of the PMD recursive loop processing for a situation where at time
k+Tp, a measurement is missing, whereby two PDF's, suffix 1 and 2,
are used for modeling, e.g. a complementary/subset model
scenario.
[0190] For both situations above it appears that since there is no
measurement at time k+1, the update leaves parameters unchanged
from the predicted values, that is,
x.sub.i.sup.u=x.sub.i.sup.p
P.sub.i.sup.u=P.sub.i.sup.p (42)
.mu..sub.i.sup.u=.mu..sub.i.sup.p
[0191] Hence, both the IMM method and the PMD method require fewer
computational steps for the update process when a measurement is
lacking.
[0192] However, when performing interaction (IMM)/re-arrangement
(PMD), a comparison of FIGS. 39 and 40 reveals that according to
the IMM method all PDF models interact in order to produce the
predicted values according to the IMM method, while no adaptation
is needed according to the PMD process according to the invention.
As can be readily understood, the computational requirements of the
invention are considerably reduced for the situation where a
measurement is lacking.
[0193] It also applies that when measurements are in fact
accomplished consecutively, the PMD process is less computational
demanding than IMM, although the effect becomes particular clear
when examining the situation for lacking measurements. An advantage
of the invention, compared to IMM, from a computational point of
view, is that the diffusion matrix M.sub.d is never a full matrix
(half of the off-diagonal elements are zero), which often is not
the case with M.sub.p used in the IMM model interaction.
[0194] In applications in which multiple predictions are made for
each measurement update, according to the invention, only one
re-arrangement is necessary.
[0195] It is noted that in typical radar applications, measurements
may occasionally not be provided for many reasons, such as low
signal/interference figures etc.
[0196] In other applications where several predictions have to be
made for each update, the same computational advantages of the
invention exist. The computational facilitation also applies for
applications where there are several measurements of which only one
(or none) should be used. Another such application is when multiple
predictions have to be made in order to establish the best time for
the measurement to be applied.
* * * * *